The starting point of our approach is the water wave system in its Hamiltonian canonical form
in terms of the surface elevation and a modified velocity potential. In Section ,
the Taylor expansion of the Hamiltonian is presented and then expressed in terms of the complex symplectic coordinates
that diagonalize its quadratic terms. Unlike the irrotational case, the linear dispersion relation is not an even function of wavenumber.
In Section 3, we introduce elements of transformation theory and Birkhoff normal form transformations.
A third-order Birkhoff normal form transformation that eliminates all non-resonant cubic terms from the Hamiltonian is explicitly calculated.
It is defined as an auxiliary Hamiltonian flow from the original variables to transformed variables.
In Section 4, we propose the new Hamiltonian truncated at fourth order
and, in Section 5, we use the modulational Ansatz together with a homogenization technique to derive the resonant quartic contributions.
In Section 6, the Hamiltonian Dysthe equation is obtained for the wave envelope.
Section 7 is devoted to a validation of our approximation through numerical simulation and comparison with other models.
A theoretical prediction on modulational stability of Stokes waves in the presence of constant vorticity is first established.
We then explain the procedure to reconstruct the surface variables by inverting the third-order Birkhoff normal form transformation,
and we describe the numerical methods to solve the equations involved in the various models.
Finally, we present a numerical investigation where predictions from our Hamiltonian Dysthe equation are compared to those from
the envelope model by [1]} and to direct simulations of the full water wave system.
